Modeling entangled dynamics: comparison between stochastic single-chain and multichain models

To test the effectiveness of stochastic single-chain models in describing the dynamics of entangled polymers, we systematically compare one such model; the slip-spring model; to a multichain model solved using stochastic molecular dynamics(MD) simulations (the Kremer-Grest model). The comparison involves investigating if the single-chain model can adequately describe both a microscopic dynamical and a macroscopic rheological quantity for a range of chain lengths. Choosing a particular chain length in the slip-spring model, the parameter values that best reproduce the mean-square displacement of a group of monomers is determined by fitting toMDdata. Using the same set of parameters we then test if the predictions of the mean-square displacements for other chain lengths agree with the MD calculations. We followed this by a comparison of the time dependent stress relaxation moduli obtained from the two models for a range of chain lengths. After identifying a limitation of the original slip-spring model in describing the static structure of the polymer chain as seen in MD, we remedy this by introducing a pairwise repulsive potential between the monomers in the chains. Poor agreement of the mean-square monomer displacements at short times can be rectified by the use of generalized Langevin equations for the dynamics and resulted in significantly improved agreement.

Stochastic climate theory and modeling

Stochastic methods are a crucial area in contemporary climate research and are increasingly being used in comprehensive weather and climate prediction models as well as reduced order climate models. Stochastic methods are used as subgrid-scale parameterizations (SSPs) as well as for model error representation, uncertainty quantification, data assimilation, and ensemble prediction. The need to use stochastic approaches in weather and climate models arises because we still cannot resolve all necessary processes and scales in comprehensive numerical weather and climate prediction models. In many practical applications one is mainly interested in the largest and potentially predictable scales and not necessarily in the small and fast scales. For instance, reduced order models can simulate and predict large-scale modes. Statistical mechanics and dynamical systems theory suggest that in reduced order models the impact of unresolved degrees of freedom can be represented by suitable combinations of deterministic and stochastic components and non-Markovian (memory) terms. Stochastic approaches in numerical weather and climate prediction models also lead to the reduction of model biases. Hence, there is a clear need for systematic stochastic approaches in weather and climate modeling. In this review, we present evidence for stochastic effects in laboratory experiments. Then we provide an overview of stochastic climate theory from an applied mathematics perspective. We also survey the current use of stochastic methods in comprehensive weather and climate prediction models and show that stochastic parameterizations have the potential to remedy many of the current biases in these comprehensive models.

Tropospheric planetary wave dynamics and mixture modeling: Two preferred regimes and a regime shift

Investigation of preferred structures of planetary wave dynamics is addressed using multivariate Gaussian mixture models. The number of components in the mixture is obtained using order statistics of the mixing proportions, hence avoiding previous difficulties related to sample sizes and independence issues. The method is first applied to a few low-order stochastic dynamical systems and data from a general circulation model. The method is next applied to winter daily 500-hPa heights from 1949 to 2003 over the Northern Hemisphere. A spatial clustering algorithm is first applied to the leading two principal components (PCs) and shows significant clustering. The clustering is particularly robust for the first half of the record and less for the second half. The mixture model is then used to identify the clusters. Two highly significant extratropical planetary-scale preferred structures are obtained within the first two to four EOF state space. The first pattern shows a Pacific-North American (PNA) pattern and a negative North Atlantic Oscillation (NAO), and the second pattern is nearly opposite to the first one. It is also observed that some subspaces show multivariate Gaussianity, compatible with linearity, whereas others show multivariate non-Gaussianity. The same analysis is also applied to two subperiods, before and after 1978, and shows a similar regime behavior, with a slight stronger support for the first subperiod. In addition a significant regime shift is also observed between the two periods as well as a change in the shape of the distribution. The patterns associated with the regime shifts reflect essentially a PNA pattern and an NAO pattern consistent with the observed global warming effect on climate and the observed shift in sea surface temperature around the mid-1970s.

Modeling of carrier transport in nanowires

We consider a physical model of ultrafast evolution of an initial electron distribution in a quantum wire. The electron evolution is described by a quantum-kinetic equation accounting for the interaction with phonons. A Monte Carlo approach has been developed for solving the equation. The corresponding Monte Carlo algorithm is NP-hard problem concerning the evolution time. To obtain solutions for long evolution times with small stochastic error we combine both variance reduction techniques and distributed computations. Grid technologies are implemented due to the large computational efforts imposed by the quantum character of the model.

(Non-)ergodicity of a degenerate diffusion modeling the fiber lay down process

We analyze the large time behavior of a stochastic model for the lay down of fibers on a moving conveyor belt in the production process of nonwovens. It is shown that under weak conditions this degenerate diffusion process has a unique invariant distribution and is even geometrically ergodic. This generalizes results from previous works [M. Grothaus and A. Klar, SIAM J. Math. Anal., 40 (2008), pp. 968–983; J. Dolbeault et al., arXiv:1201.2156] concerning the case of a stationary conveyor belt, in which the situation of a moving conveyor belt has been left open.

Analysis and modeling of active occupancy of the residential sector in Spain: an indicator of residential electricity consumption

The growing energy consumption in the residential sector represents about 30% of global demand. This calls for Demand Side Management solutions propelling change in behaviors of end consumers, with the aim to reduce overall consumption as well as shift it to periods in which demand is lower and where the cost of generating energy is lower. Demand Side Management solutions require detailed knowledge about the patterns of energy consumption. The profile of electricity demand in the residential sector is highly correlated with the time of active occupancy of the dwellings; therefore in this study the occupancy patterns in Spanish properties was determined using the 2009–2010 Time Use Survey (TUS), conducted by the National Statistical Institute of Spain. The survey identifies three peaks in active occupancy, which coincide with morning, noon and evening. This information has been used to input into a stochastic model which generates active occupancy profiles of dwellings, with the aim to simulate domestic electricity consumption. TUS data were also used to identify which appliance-related activities could be considered for Demand Side Management solutions during the three peaks of occupancy.

Ensemble downscaling in coupled solar wind-magnetosphere modeling for space weather forecasting

Advanced forecasting of space weather requires simulation of the whole Sun-to-Earth system, which necessitates driving magnetospheric models with the outputs from solar wind models. This presents a fundamental difficulty, as the magnetosphere is sensitive to both large-scale solar wind structures, which can be captured by solar wind models, and small-scale solar wind “noise,” which is far below typical solar wind model resolution and results primarily from stochastic processes. Following similar approaches in terrestrial climate modeling, we propose statistical “downscaling” of solar wind model results prior to their use as input to a magnetospheric model. As magnetospheric response can be highly nonlinear, this is preferable to downscaling the results of magnetospheric modeling. To demonstrate the benefit of this approach, we first approximate solar wind model output by smoothing solar wind observations with an 8 h filter, then add small-scale structure back in through the addition of random noise with the observed spectral characteristics. Here we use a very simple parameterization of noise based upon the observed probability distribution functions of solar wind parameters, but more sophisticated methods will be developed in the future. An ensemble of results from the simple downscaling scheme are tested using a model-independent method and shown to add value to the magnetospheric forecast, both improving the best estimate and quantifying the uncertainty. We suggest a number of features desirable in an operational solar wind downscaling scheme.